Math and stuff

Proposition Let $a$ be an ideal of a ring $A$, and let $S = 1 + a$. Show that $S^{-1}a$ is contained in the Jacobson radical of $S^{-1}A$. Use this result and Nakayama’s lemma to give a proof of (2.5)[Atiyah] which does not depend on determinants. Solution $0 \in a$,...

Proposition Suppose that $f$ has a zero of multiplicity $m$ at $a$. Explain why $1/f$ has a pole of order $m$ at $a$. Solution $f$ has a zero of multiplicity $m$ at $a$. By Theorem 8.14[A first course in complex analysis], this means that there exists a holomorphic function $g$...

Proposition Find the poles or removable singularities of the following functions and determine their orders: $(z^2 + 1)^{-3}(z - 1)^{-4}$. $z\cot(z)$. $z^{-5}\sin z$. $\frac{z}{1 - \exp(z)}$. $\frac{1}{1 - \exp(z)}$. Solution 1 The order of the pole at $i$ is $3$. The order of the pole at $1$ is $4$. 2...

Proposition Show that if $f$ has an essential singularity at $z_0$ then $\frac{1}{f}$ also has an essential singularity at $z_0$. Solution Suppose $1/f$ does not have an essential singularity at $z_0$. Since $f$ has an isolated singularity at $z_0$, $1/f$ has an isolated singularity at $z_0$. By Proposition 9.5[a first...

Proposition If $w_1, \cdots, w_{n - 1} \in \mathbb{R}^n$, show that \[\begin{align*} \abs{w_1 \times \cdots \times w_{n - 1}} = \sqrt{\det(g_{ij})}, \end{align*}\] where $g_{ij} = \ev{w_i, w_j}$. Solution Suppose $w_1, \cdots, w_{n - 1}$ are linearly dependent. Then $w_1 \times \cdots \times w_{n - 1} = 0$ because $\ev{w_1 \times...